CN111882156B - Train schedule robust optimization method for random dynamic passenger flow and energy-saving operation - Google Patents
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Abstract
The invention relates to a robust optimization method for train timetables facing to random dynamic passenger flow and energy-saving operation, which comprises the following steps: s1, introducing an effective service decision variable, a departure time decision variable of a train at a first station and an operation curve selection decision variable among stations into a robust optimization model of a train schedule; s2, constructing a linear association constraint between a train schedule, a departure time decision variable of the train and an inter-station operation curve selection decision variable; s3, constructing nonlinear association constraint between the train schedule and the effective service decision variables; constructing train operation safety time interval constraint and train bearing capacity constraint; s4, combining the steps S1-S3 to construct a robust optimization model of the train schedule for random dynamic passenger flow and energy-saving operation; and S5, solving the robust optimization model to obtain an optimal train schedule. The method can efficiently solve the subway train schedule optimization problem under the uncertain passenger flow condition, and improve the robustness of knowing.
Description
Technical Field
The invention belongs to the technical field of urban rail transit transportation organizations, relates to a subway train schedule robust optimization method for random dynamic passenger flow and energy-saving operation, and particularly relates to a subway train schedule robust optimization method under uncertain passenger flow conditions.
Background
Various traffic modes in the large city have the characteristics, and the rail transit is taken as the main force and backbone of public transportation, so that the method is a solution to solve the problems of traffic jam, environmental pollution, resource waste and the like. Urban rail transit bears large-scale travel demands by virtue of the characteristics of convenience, punctuality, large traffic, low emission, high reliability and the like. The urban rail transit requirement has the basic characteristics of complexity, space-time diversity and the like, presents high uncertainty in the daily operation process, and the development of the urban rail transit is limited by factors such as train operation speed, passenger flow and the like. Therefore, a subway train robust schedule adapting to uncertain passenger flow demands is compiled from a system optimization level so as to reasonably distribute the operation capacity and improve the operation efficiency, and the method is a problem to be solved in urban rail transit transportation organizations.
Train schedules occupy an extremely important place throughout urban rail transit systems. On the one hand, the method is a basic file for ensuring the safety and normal operation of the train, and is an effective expression form for providing service for society. As an important link in train operation planning and the basis of subway company organization train operation, the train schedule optimization design problem is that on the basis of a given line, after the number of train groups is determined, the departure density, the running time and the stop time are determined according to the passenger flow requirement and the number of train bottoms, and the arrival and departure time of each train at each station is determined. Currently, researches on schedule optimization aiming at the technical means of the number of established trains, the adjustment of stop time, running time and the like and considering deterministic passenger flow demands have been accumulated to a certain extent. However, there are relatively few studies on optimizing designs of train schedules based on random dynamic passenger flow demands while taking into account operational company, passenger and social benefits (i.e., operating costs, trip time and energy-saving operation).
At present, although many scholars have studied the optimization method of train schedules, most of them are based on the given number of trains and the running time of the trains between stations. From the system optimization perspective, this hierarchical decision method generally only can obtain a locally optimal solution of two sub-problems, and cannot fundamentally meet the goal of providing passengers with higher service levels at lower cost. For example, in peak time, the passenger flow is large, and 12 trains need to be started within 1 hour at 5 minutes as time intervals; in the flat peak period, the passenger flow is smaller, and only 10 trains need to be started when 6 minutes are taken as time intervals; and at night, the passenger flow is smaller, and only 6 trains need to be started at intervals of 10 minutes. Therefore, under the condition of larger passenger flow, the former needs higher running cost, but can effectively reduce the waiting time of passengers to improve the service level; and under the condition of smaller passenger flow, on the premise of ensuring the waiting time of passengers, the main aims of reducing the operation cost and improving the economic benefit are realized. In addition, the running time of the train between stations and the total mass of the train as it runs between stations also have a direct impact on the magnitude of the train traction energy consumption and auxiliary energy consumption. Therefore, under the condition of uncertain passenger flow, a robust optimization method for the train schedule oriented to random dynamic passenger flow and energy-saving operation needs to be provided, and the number of running trains and the corresponding train schedule are optimized at the same time from the system optimization level so as to obtain a system optimal rail transit organization scheme and provide decision support for realizing refined operation of urban rail transit.
Disclosure of Invention
The invention aims to provide a robust optimization method of a train schedule for random dynamic passenger flow and energy-saving operation, which aims to solve the problems that the existing method only considers the number of running trains, the running curve between stations and the train schedule to generate local optimal solutions in a stepwise independent mode under the deterministic passenger flow condition, and solves the complex problems of flat peak and peak schedules for many times according to time periods, thereby improving the system optimization quality of the subway train schedule and avoiding the important problems of excessive train capacity, excessively high traction energy consumption, excessively long waiting time of passengers and the like.
In order to achieve the purpose, the method comprises the steps of discretizing a planning time interval into a plurality of time periods, adopting a scene-based time-dependent OD passenger flow matrix, establishing a robust optimization model of subway train timetable, optimizing the departure interval of each train at the first station and the running curve between each station, and improving the robustness of solutions. Finally, the model is effectively solved by adopting optimization software (CPLEX) and Variable Neighborhood Search (VNS) algorithms respectively. The specific technical scheme is as follows:
A robust optimization method for train timetable facing random dynamic passenger flow and energy-saving operation comprises the following steps:
s1, recording station set of lines in a route as The station 1 is a starting station, and the station N is a destination station; record the set of available physical trains as/>A total of K trains can be put into service; recording the scene set of passenger flow as/>Record the discrete time period set/>Wherein t represents a t-th period;
Effective service decision variable y k, train departure time decision variable d 1,k at the first station and running curve selection decision variable among stations are introduced into robust optimization model of train schedule Where k represents train k,/>
S2, analyzing the relation between the train schedule and the departure time decision variable of the train at the head station and the operation curve selection decision variable among stations, and constructing linear association constraint between the train schedule and the departure time decision variable d i,k of the train and the operation curve selection decision variable among the stations;
s3, constructing nonlinear association constraint between the train schedule and an effective service decision variable y k; constructing train operation safety time interval constraint and train bearing capacity constraint;
S4, constructing a robust optimization model of a train schedule for random dynamic passenger flow and energy-saving operation, which aims at the fixed cost of the train, the total waiting time of passengers and the total energy consumption of the train, by combining the steps S1-S3, wherein decision variables of the robust optimization model comprise effective service decision variables, decision variables of departure time of the train at the departure time of the first station and decision variables of operation curves of the train among stations;
and S5, solving the robust optimization model to obtain a global optimal solution serving as an optimal train schedule.
In step S1, the valid service decision variable y k =1 indicates that the train k performs a valid service, and y k =0 indicates that the train k performs an invalid service;
The departure time decision variable d 1,k of the train at the first stop represents the moment when the train k leaves the station 1;
Decision variable for inter-station operation curve selection The selection curve f is shown when the train k is operating in the interval i, i+1.
The specific steps of the step S2 are as follows: introducing a binary auxiliary variable z i,k(t),zi,k (t) =1 to indicate that the train k has arrived or passed the station i at the time t, and z i,k (t) =0 to indicate that the train k has not arrived and has not passed the station i at the time t, wherein the variable satisfies a non-increasing constraint as shown in the formula (2); the departure time d i,k of the train k at the station i is expressed as the expression (3):
Wherein, For station set in route,/>For the set of available physical trains,/>Is a set of discrete time periods; d i,k is a departure time decision variable of the train, which indicates that the train k starts from the station i at the time d i,k, a i+1,k indicates that the train arrives at the subsequent station i+1 at the time a i+1,k, and the running time r i,k of the interval [ i, i+1] is determined by the running curve selected when the train k runs on the interval; at this time, there is a linear association constraint between the train schedule and the departure time decision variable and the inter-station operation curve selection decision variable of the train as follows:
Wherein, For station set in route,/>For the set of available physical trains, a i,k is the time when train k arrives at station i, d i,k is the time when train k leaves station i, s i,k is the stop time of train k at station i, r i,k is the running time of train k in interval [ i, i+1 ]/>For all the set of alternative operating curves over interval [ i, i+1 ]/>The operation time corresponding to the operation curve f in the section [ i, i+1 ]; the formula (4) and the formula (5) are the departure and arrival time constraints of the train, the formula (6) is the association constraint of the decision variables selected by the running time between stations and the running curve between stations, and the formula (7) is the unique constraint of the running curve selected by the train k when running on the interval [ i, i+1 ].
The step of constructing a nonlinear association constraint between the train schedule and the effective service decision variable y k in step S3 is: establishing a non-subtractive constraint for the effective service decision variable y k as shown in equation (8):
in order for a train executing effective service to complete the service within a planned time interval, the departure time of an effective terminal at each station should be later than the arrival time boundary of the passenger at the station, and the departure time of an effective next terminal at each station should be strictly earlier than the arrival time boundary of the passenger at the station; if train Executing effective service, recording y k =1, and adopting y k-yk+1 =1 to represent that the train k is an effective terminal bus, and adopting y k+1-yk+2 =1 to represent that the train k is an effective next terminal bus; the departure time constraint of the effective terminal bus at each station and the departure time constraint of the effective sub terminal bus at each station are respectively shown as the following formula (9) and the formula (10):
Wherein, A passenger arrival time boundary for station i; the formulas (9) and (10) are nonlinear constraints.
The step of constructing the train operation safety time interval constraint in the step S3 is as follows: the departure time interval of the train k and the preceding train k-1 at the station i is recorded as h i,k,hi,k＝di,k-di,k-1,The train safety time interval constraint is constructed as follows:
Wherein H and The minimum and maximum interval time for two adjacent trains to leave or arrive at the station are respectively.
The step of constructing the bearing capacity constraint of the train in the step S3 is as follows: constructing a binary auxiliary variable L i,k(t),Li,k (t) =1 representing that a passenger of the station i can ride the train number k in the current t-th time period by the binary auxiliary variable z i,k (t) representing the train state in the step S2, wherein the t-th time period represents that the passenger reaching the station i can ride the train number k; l i,k (t) =0, indicating that the passenger arriving at station i in the t-th time period cannot ride the number k of cars; specifically expressed as formula (16):
Therefore, the number of passengers getting on and off the train k at the station i, the number of passengers in the carriage and the constraint of the train bearing capacity under the omega scene are constructed as follows:
Wherein, P i,j,ω (t) is the number of passengers arriving at station i and going to station j in the t-th time period under the omega scene, and P i′,i,ω (t) is the number of passengers arriving at station i' and going to station i in the t-th time period under the omega scene; in formula (17) For the number of boarding persons for train k at station i in ω scene,/>, in equation (18)For the number of passengers getting off the train k at the station i in the omega scene, C i,k,ω is the total number of passengers in a carriage when the train k leaves the station i in the omega scene, and C max is the maximum bearing capacity of the train; equation (19) describes the dynamic change in the number of passengers in station i car for train k in ω scenario, which must meet the maximum capacity constraint (20) of the train to guarantee the feasibility of the current train schedule.
The specific steps of the step S4 are as follows: (1) The total number of effective train numbers running in the planning time interval isAccording to the effective service unit fixed cost Q 0, an objective function representing the total operation cost is established, as shown in the formula (21), and the total fixed cost of the train is as follows:
(2) After the passengers enter the platform, waiting for the incoming trains in the waiting area, wherein the waiting time of the passengers which arrive at the station i in the t-th time period and leave by taking the train number k is d i,k -t, and according to the definition of the binary auxiliary variable L i,k (t), the expected value of the total waiting time of the passengers is calculated by the following formula:
Where TW ω is the total waiting time of the passengers in the ω scene and α ω is the probability of scene ω.
(3) The total energy consumption of the train comprises: traction energy consumption related to total mass of the train and auxiliary energy consumption related to the number of passengers and the inter-station travel time; the mass of the train in the empty state is recorded as M 0, and the average mass of passengers is recorded as M p; the unit traction energy consumption is u i,k when the train runs in an empty state, and is specifically expressed as a formula (26); traction energy consumption occurs if and only if the train performs an active service, and thus the traction energy consumption when the train k runs over the section [ i, i+1] in ω scene is expressed as the formula (27):
Wherein mu 1,μ2,μ3 is a constant parameter calculated from actual operation data;
Note that P ac is the average power consumption per unit time in the carriage, and P lc is the power consumption per unit time of auxiliary equipment in the carriage; likewise, auxiliary energy consumption occurs if and only if the train performs an active service, and thus, the auxiliary energy consumption when the train k runs on the section [ i, i+1] in the ω scene is represented by the formula (28):
then, the total energy consumed by the train running in ω scene is calculated by the equation (29):
Thus, the expected value of the total energy consumption of a scene-based train is:
the train schedule robust optimization model for random dynamic passenger flow and energy-saving operation constructed by the invention is as follows:
j is the total cost of the operator and the passenger in the planned time interval; FC is the total fixed cost of the train; TW ω is the total waiting time of the passengers in scene ω; e ω is the total energy consumption of the train in scene omega; alpha ω is scene probability; beta c,βt,βe is the weight coefficient of the fixed cost, the passenger waiting time cost and the train energy consumption cost, respectively.
In step S5, solving the robust optimization model by adopting a variable neighborhood search algorithm, wherein the method comprises the following specific steps: setting dithering operatorsI.e. randomly generating a new feasible solution, wherein/>In the form of a vector of decision variables y k,/>In the form of a vector of decision variables d 1,k; definition of neighborhood Structure/>Namely exchanging departure intervals of the train number k 1 and the preceding train k 1 -1 and the train number k 2 and the preceding train k 2 -1, wherein k 1,k2 is a randomly selected valid service train number; definition of neighborhood Structure/>I.e. changing the inter-station operating curve of each train number, where/>For decision variables/>Vector form of (a); definition of neighborhood Structure/>Regenerating a feasible departure time sequence/>, according to the number of effective service vehicles K p in the current solutionSelecting a feasible running curve for each train; thus, the form of the solution of the constructed robust optimization model can be expressed as/>Recording search times count=0 of failing to find a better solution, wherein the termination condition is that count is more than or equal to MAX iteration;
step 1, generating an initial feasible solution as S 0, and enabling a global optimal solution S=S 0;
step2, perturbing the global optimal solution S according to the dithering operator M to generate an initial local optimal solution S' =m (S);
Step 3, performing neighborhood search on the solution S ' by using a neighborhood structure N 1, and if a neighbor solution S "=n 1 (S ') is found and f (S") < f (S '), updating the locally optimal solution: s' =s ", and repeat Step 3; otherwise, turning to Step 4;
Step 4, performing neighborhood search on the solution S ' by using a neighborhood structure N 2, and if a neighbor solution S "=n 2 (S ') is found and S (S") < f (S '), updating the locally optimal solution: s' =s ", and repeat Step3; otherwise, turning to Step 5;
Step 5, performing neighborhood search on the solution S ' by using a neighborhood structure N 3, and if a neighbor solution S "=n 3 (S ') is found and f (S") < f (S '), updating the locally optimal solution: s' =s ", and repeat Step3; otherwise, turning to Step 6;
Step 6, comparing the local optimal solution S 'with the global optimal solution S, and if f (S') < f (S), updating the global optimal solution: s=s', and let count=0, go to Step 2; otherwise, let count=count+1, go to Step 7;
Step 7, if the termination condition count is not more than MAX iteration, turning to Step 2; otherwise, the algorithm is terminated, and a global optimal solution S is output.
The invention has the beneficial effects that:
The technical scheme of the invention realizes the system optimization of the train schedule and the train operation curve, and has the following advantages: (1) Meanwhile, the running quantity of trains and running time among stations are optimized, and corresponding train schedules are improved, so that the system optimization quality of subway train schedules is improved obviously; (2) The robustness of the train schedule is effectively improved by utilizing random dynamic passenger flow data based on scenes; (3) All constraint conditions can be converted into linear forms, and the solution can be carried out by utilizing general commercial optimization software, so that the complexity of the model is reduced; (4) The designed variable neighborhood search algorithm can greatly shorten the solving time and effectively improve the solving efficiency on the premise of ensuring the solving quality; (5) The model constructed by the method is accurate, simple in form and good in expansion space, so that some more complex practical applications can be met.
Drawings
The invention has the following drawings:
Fig. 1 shows an algorithm flow diagram of a variant neighborhood search algorithm.
Fig. 2 shows a single line subway line map.
Fig. 3 shows a distribution diagram of the passenger arrival rate over time.
Fig. 4 shows a train operation diagram at peak hours.
Fig. 5 shows a train operation diagram at peak flattening time.
Fig. 6 shows a night time train operation diagram.
Fig. 7 shows the departure time intervals of trains from the head station for different time periods.
Detailed Description
In order to more clearly illustrate the present invention, the present invention will be further described with reference to preferred embodiments and the accompanying drawings. It is to be understood by persons skilled in the art that the following detailed description is illustrative and not restrictive, and that this invention is not limited to the details given herein.
Generally, in the train schedule optimization problem, decision variables are typically represented by the time points a i,k and d i,k representing the arrival and departure of train k at station i, the number of trains required in the planned time frame is typically calculated by summing the effective service decision variables y k, and the running time and energy consumption of the trains between stations are selected by running curves to decide the variablesTo calculate. The following links exist between the decision variables: if the train k is an effective service, i.e. y k =1, the departure time of the effective terminal at each station should be later than the arrival time boundary of the passenger, and the departure time of the effective terminal at each station should be strictly earlier than the arrival time boundary of the passenger, the effective service will correspond to fixed operation cost and effective traction and auxiliary energy consumption. Furthermore, the goal of efficient service and train schedule optimization is to provide passengers with efficient and high quality service at a lower cost. Therefore, the consistency of the optimization targets and the relevance between the decision variables ensure the feasibility of constructing a robust optimization method for train schedules facing random dynamic passenger flows and energy-saving operation.
Based on the analysis, the subway train schedule robust optimization method facing random dynamic passenger flow and energy-saving operation comprises the following steps:
The station set of the line in the recording route is Wherein station 1 is the originating station and station N is the terminal station. Record the set of available physical trains as/>I.e. a total of K trains can be serviced. The planned time interval is noted as [ T s,Te ], and if the train k can complete the passenger transportation service from the origination station to the destination station within the planned time interval, it is called an effective service. Recording the scene set of passenger flow as/>Firstly, dividing a planning time interval [ T s,Te ] into T discrete time segments by using a time granularity with the length delta, and recording a discrete time segment set/>Where t represents the t-th period. Scene/>, using a time-dependent OD matrix P ω (t)The passenger travel demand reached in the next t-th time period is specifically shown as a formula (1),
Wherein P i,j,ω (t) represents the number of passengers from station i to station j in the t-th time period under scene ω. By means of the method, the travel demands of passengers in the planning time interval can be dynamically represented.
S1, introducing an effective service decision variable, a departure time decision variable of the train and an operation curve selection decision variable into a train schedule optimization model. Thus, the decision variables of the train schedule optimization model include:
The effective service decision variable y k,yk =1 represents a train Executing an active service, y k =0 represents train/>Executing invalid service;
The departure time decision variable d 1,k,d1,k of the train at the first stop represents the moment when the train k leaves the station 1;
Decision variable for inter-station operation curve selection The selection curve f is shown when the train k is operating in the interval i, i+1.
S2, analyzing the relation between the train schedule and the decision variable of the departure time of the train at the head station and the decision variable of the operation curve selection between stations, and constructing the train schedule and the decision variable d i,k of the departure time of the train and the decision variable of the operation curve selection between stationsThe linear association constraint between the two is as follows:
To build a linear programming model that is convenient to solve, a binary auxiliary variable Z i,k(t),Zi,k (t) =1 is introduced to indicate that the train k has arrived at or passed the station i at the time t, Z i,k (t) =0 indicates that the train k has not arrived at and has not passed the station i at the time t, and the variable satisfies a non-increasing constraint, as shown in formula (2). The departure time d i,k of the train k at the station i can be expressed as the expression (3):
Wherein, For station set in route,/>For the set of available physical trains,/>Is a set of discrete time periods. d i,k is a departure time decision variable of the train, which indicates that the train k starts from the station i at the time d i,k and reaches the subsequent station i+1 at the time a i+1,k, and the running time r i,k of the section [ i, i+1] is determined by the running curve selected when the train k runs on the section. At this time, there is a linear association constraint between the train schedule and the departure time decision variable and the inter-station operation curve selection decision variable of the train as follows:
Wherein, For station set in route,/>For the set of available physical trains, a i,k is the time when train k arrives at station i, d i,k is the time when train k leaves station i, s i,k is the stop time of train k at station i, r i,k is the running time of train k in interval [ i, i+1 ]/>For all the set of alternative operating curves over interval [ i, i+1 ]/>Is the run time corresponding to the run curve f within section i, i+1. Equation (4) and equation (5) are train arrival time constraints, equation (6) is an association constraint of a decision variable selected for the inter-station operation time and the inter-station operation curve, and equation (7) is a unique constraint of the selected operation curve when the train k operates on the interval [ i, i+1 ].
S3, analyzing the relation between the train schedule and the effective service decision variable, and constructing nonlinear association constraint between the train schedule and the effective service decision variable, wherein the specific process is as follows:
a non-subtractive constraint is established for the effective service decision variable y k as shown in equation (8).
In order for a train executing an effective service to be able to complete the service within a planned time interval, the departure time of the effective terminal at each station should be later than the arrival time boundary of the passenger at that station, and the departure time of the effective next terminal at each station should be strictly earlier than the arrival time boundary of the passenger at that station. If trainEffective service is performed, note y k =1, and let y k-yk+1 =1 denote train k as the effective terminal, and y k+1-yk+2 =1 denote train k as the effective next terminal. The departure time constraint of the effective terminal bus at each station and the departure time constraint of the effective sub terminal bus at each station are respectively shown as the following formula (9) and the formula (10):
Wherein, Is the passenger arrival time boundary for station i. Since d i,k and y k are both decision variables, the above formulas (9) and (10) are nonlinear constraints, which can be converted into linear constraints for easy solution by the following method:
Recording device Then/>Representing that the train k is an effective terminal bus; the departure time of the effective terminal bus at each station is recorded as/>And is expressed as/>Wherein due to/>Represented by decision variables, therefore,/>As a nonlinear constraint, it can be converted into the following linear form by introducing a sufficiently large positive integer U, as shown in formula (11):
Thus, the nonlinear constraint equation (9) can be converted into the following linear form, as shown in equation (12):
similarly, record Then/>Representing that the train k is a valid last shift; the departure time of the effective next-time terminal bus at each station is recorded as/>And is expressed as/>Wherein due to/>Represented by decision variables, thus/>For non-linear constraint, a positive integer U can be introduced that is sufficiently large to convert it to the following linear form, as shown in equation (13):
thus, the nonlinear constraint equation (10) can be converted into the following linear form, as shown in equation (14):
Besides the linear association constraint between the train schedule and the departure time decision variable of the train and the inter-station operation curve selection decision variable and the nonlinear association constraint between the train schedule and the effective service decision variable, the train operation safety time interval constraint and the train bearing capacity constraint are constructed, and the method is specifically as follows:
(1) Safety time interval constraint:
In an urban rail transit system, in order to ensure the safety of trains in the running process and avoid overlong waiting time of passengers, a certain time interval must be kept between two adjacent trains. The departure time interval of the train k and the preceding train k-1 at the station i is recorded as h i,k, namely h i,k＝di,k-di,k-1, Thus, the train safety interval constraint may be constructed as follows:
Wherein H and The minimum and maximum interval time for two adjacent trains to leave or arrive at the station are respectively.
(2) Train bearing capacity constraints:
In the dynamic loading process of passenger flow, the bearing capacity of the train at each station is an important index, which is continuously changed along with the process of getting on and off each station, so that the total passenger carrying quantity of each train does not exceed the maximum bearing capacity of the train in any time period, and the binary auxiliary variable z i,k (t) representing the state of the train in the step S2 is used for constructing the binary auxiliary variable L i,k(t),Li,k (t) =1 representing the passenger capable of taking the passenger k of the station i in the current t time period to represent the passenger capable of taking the passenger k of the station i in the t time period; l i,k (t) =0, indicating that the passenger arriving at station i in the t-th period cannot ride the number k of cars. Specifically expressed as formula (16):
Therefore, the number of passengers getting on and off the train k at the station i, the number of passengers in the carriage and the constraint of the train bearing capacity under the omega scene are constructed as follows:
Wherein, P i,j,ω (t) is the number of passengers arriving at station i and going to station j in the t-th time period under the omega scene, and P i′,i,ω (t) is the number of passengers arriving at station i' and going to station i in the t-th time period under the omega scene. In formula (17) For the number of boarding persons for train k at station i in ω scene,/>, in equation (18)For the number of passengers getting off the train k at the station i in the omega scene, C i,k,ω is the total number of passengers in the carriage when the train k leaves the station i in the omega scene, and C max is the maximum bearing capacity of the train. Equation (19) describes the dynamic change in the number of passengers in station i car for train k in ω scenario, which must meet the maximum capacity constraint (20) of the train to guarantee the feasibility of the current train schedule.
S4, constructing a robust optimization model of a train schedule for random dynamic passenger flow and energy-saving operation by taking train fixed cost, train total energy consumption and total waiting time of passengers as optimization targets by combining the steps S1-S3, wherein decision variables of the robust optimization model comprise effective service decision variables, train departure time decision variables and inter-station operation curve selection decision variables, and the specific process is as follows:
The main purpose of urban rail transit organization optimization is to provide high quality service for passengers with the lowest cost, and the stakeholders thereof mainly include passengers and subway operation companies. Passengers desire that subway operators can provide high quality, convenient, fast transportation services, while subway operators desire to provide these necessary services at lower costs. Therefore, passenger total waiting time, total travel time, comfort, total train operation cost, total energy consumption, etc. are often used as important indicators for evaluating the quality of subway transportation organization schemes. The steps of the train schedule robust optimization model constructed by the invention and oriented to random dynamic passenger flow and energy-saving operation are as follows:
(1) As can be seen from the definition of the decision variables, the total number of effective train numbers running in the planning time interval is Based on the effective service unit fixed cost Q 0, an objective function representing the total operation cost is established as shown in formula (21), that is, the total fixed cost of the train (which is the fixed cost of all trains performing effective service) is:
(2) After passengers enter the platform, the waiting area waits for an incoming train, so that the passenger waiting time is closely related to the train running pattern and the passenger capacity. The passenger waiting time for the t-th time period to reach the station i and leave by the number of passengers k is (d i,k -t), and according to the definition of the binary auxiliary variable L i,k (t), the expected value of the total waiting time of the passengers can be calculated by the following formula:
Where TW ω is the total waiting time of the passengers in the ω scene and α ω is the probability of scene ω. Since both L i,k (t) and d i,k in formula (23) contain decision variables, formula (23) has nonlinear properties. For ease of solution, it can be converted into a linear equation by:
Recording device Representing the total time for a passenger in ω scene to wait for train k from station i in the t-th time period, also of a nonlinear nature, it can be transformed into the following linear equation:
Thus, formula (23) may be converted into:
(3) During daily operation, the total train energy consumption is usually composed of two parts: traction energy consumption related to the total mass of the train and auxiliary energy consumption related to the number of passengers and the inter-station travel time. The mass of the train in the empty state is recorded as M 0, and the average mass of passengers is recorded as M p. The unit traction energy consumption is u i,k when the train runs in an empty state, and is specifically expressed as a formula (26). Traction energy consumption occurs if and only if the train performs an active service, and thus, traction energy consumption when the train k runs on the section [ i, i+1] in ω scene can be expressed as formula (27):
/>
Wherein μ 1,μ2,μ3 is a constant parameter calculated from actual operating data.
Note that P ac is the power consumption per unit time in the vehicle, and P lc is the power consumption per unit time in the auxiliary equipment in the vehicle. Likewise, auxiliary energy consumption occurs if and only if the train performs an active service, and thus, auxiliary energy consumption when the train k runs on the section [ i, i+1] in ω scene can be expressed as the expression (28):
the total energy consumed by the train operation in ω scenarios can be calculated by the equation (29):
Thus, the expected value of the total energy consumption of a scene-based train is:
Since C i,k,ω and u i,k in equation (27) are represented by decision variables d i,k and u i,k, respectively The equation (27) is calculated to contain the decision variable y k and thus has nonlinear properties. Record/>Indicating that train k is performing active service and selecting curve f while running in interval i, i + 1. y k and/>Are all 0-1 variables, thus/>The following constraints are satisfied:
Recording device For train k to run in interval [ i, i+1] and select curve/>The traction energy consumption can be calculated by the following linear relation:
wherein U is a sufficiently large positive integer. Thus, formula (27) may be expressed as:
Similarly, record For train k to run in interval [ i, i+1] and select curve/>Energy consumption at that time, equation (28) can be converted to a linear form of equation (35):
/>
the train schedule robust optimization nonlinear programming model for random dynamic passenger flow and energy-saving operation constructed by the invention is as follows:
j is the total cost of the operator and the passenger in the planned time interval; FC is the total fixed cost of the train; TW ω is the total waiting time of the passengers in scene ω; e ω is the total energy consumption of the train in scene omega; alpha ω is scene probability; beta c,βt,βe is the weight coefficient of the fixed cost, the passenger waiting time cost and the train energy consumption cost, respectively.
To facilitate solution with commercial software CPLEX, etc., the nonlinear programming model (36) can be converted into the following robust optimized linear programming model for train schedules for random dynamic passenger flow and energy-efficient operation:
s5, effectively solving the robust optimization model by adopting a variable neighborhood search algorithm. The main ideas of the variable neighborhood search algorithm are as follows: and setting a plurality of different neighborhoods for the initial feasible solution to perform system search. First a minimum neighborhood search is employed. When the current solution cannot be improved, switching to a slightly larger neighborhood to continue searching; if the current solution can be improved, the method returns to the minimum neighborhood, otherwise, the method continues to switch to a larger neighborhood. The step of solving the robust optimization problem of the train schedule facing the random dynamic passenger flow and energy-saving operation by using the variable neighborhood search algorithm can be described as follows:
Setting dithering operators I.e. randomly generating a new feasible solution, wherein/>In the form of a vector of decision variables y k,/>In the form of a vector of decision variables d 1k; definition of neighborhood Structure/>Namely exchanging departure intervals of the train number k 1 and the preceding train k 1 -1 and the train number k 2 and the preceding train k 2 -1, wherein k 1,k2 is a randomly selected valid service train number; definition of neighborhood Structure/>I.e. changing the inter-station operating curve of each train number, where/>For decision variables/>Vector form of (a); definition of neighborhood Structure/> Regenerating a feasible departure time sequence/>, according to the number of effective service vehicles K p in the current solutionAnd selects a viable operating curve for each train. Thus, the form of the solution of the constructed robust optimization model can be expressed as/>The search times of the better solution can not be found out, count=0, and the termination condition is that count is not less than MAX iteration.
Step 1, generating an initial feasible solution as S 0, and enabling a global optimal solution S=S 0;
step2, perturbing the global optimal solution S according to the dithering operator M to generate an initial local optimal solution S' =m (S);
Step 3, performing neighborhood search on the solution S ' by using a neighborhood structure N 1, and if a neighbor solution S "=n 1 (S ') is found and f (S") < f (S '), updating the locally optimal solution: s' =s ", and Step 3 is repeated. Otherwise, turning to Step 4;
Step 4, performing neighborhood search on the solution S ' by using a neighborhood structure N 2, and if a neighbor solution S "=n 2 (S ') is found and f (S") < f (S '), updating the locally optimal solution: s' =s ", and Step3 is repeated. Otherwise, turning to Step 5;
step 5, performing neighborhood search on the solution S ' by using a neighborhood structure N 3, and if a neighbor solution S "=n 3 (S ') is found and f (S") < f (S '), updating the locally optimal solution: s' =s ", and Step3 is repeated. Otherwise, turning to Step 6;
Step 6, comparing the local optimal solution S 'with the global optimal solution S, and if f (S') < f (S), updating the global optimal solution: s=s', and let count=0, go to Step 2. Otherwise, let count=count+1, go to Step 7.
Step 7, if the termination condition count is not more than MAX iteration, turning to Step 2. Otherwise, the algorithm is terminated, and a global optimal solution S is output.
A variable neighborhood search algorithm flow chart is shown in fig. 1.
Example analysis
The subway train schedule robust optimization model for random dynamic passenger flow and energy-saving operation constructed by the invention mainly comprises a binary variable y k from the aspect of the type of decision variable,And a non-negative integer variable d 1,k, the complexity of the model is primarily dependent on the number of trains, stops, and alternate curves of operation between stops. In addition, the constructed constraint conditions can be converted into linear constraint, so that common commercial optimization software (such as CPLEX, GUROBI and the like) can be utilized to solve a model with a general scale, and a train schedule with a better system is obtained. However, when the number of stations in the line is large or the optimization time period is long, the problem scale is greatly increased, and the problem is difficult to solve by using optimization software, and at the moment, the variable neighborhood search algorithm can greatly improve the solving efficiency in a limited time, so that an approximate optimal solution is obtained.
The robust optimization method for the train schedule for random dynamic passenger flow and energy-saving operation disclosed by the invention is further described below by taking a single-line subway line diagram as shown in fig. 2 as an example.
First, the following necessary parameters and data are predetermined:
(1) Station set in route Inter-station zone set/>Available physical train set/>Set of train operation curves in section [ i, i+1 ]/>Discrete time set/>Scene set/>Respectively selecting dynamic passenger flow data of different time periods (Gao Fengyi and a flat peak period) as input, wherein the representation form of the dynamic passenger flow data is an OD matrix shown in a formula (1), and the distribution of the passenger arrival rate along with time is shown in fig. 3;
(2) Minimum and maximum time interval threshold values H and H for two adjacent trains to arrive at or leave the same station Train maximum bearing capacity C max, train stop time s i,k;
(3) Train mass M 0, average passenger mass M p, passenger average energy consumption P ac in a carriage and energy consumption P lc of auxiliary equipment (air conditioner, illumination and the like) in the carriage;
(4) Unit fixed cost (employee wages, equipment losses, etc.) Q 0 for each effective service performed by the train;
(5) Run time corresponding to run curve f i within section [ i, i+1] Energy consumption per unit system/>
(6) Scene probability α ω, fixed cost, passenger waiting time cost, and weight coefficient β c,βt,βe of train energy consumption cost.
Secondly, under the condition that the conditions are given, codes are written, a model framework proposed by the method is constructed, optimization software CPLEX is respectively called in MATLAB to carry out solving, a variable neighborhood search algorithm is designed to carry out solving, a corresponding train schedule is obtained, and the results are analyzed and compared.
The specific process is as follows:
according to the single line subway line map as shown in fig. 2, the following known parameters and data are given:
(1) The line comprises 4 stations and 3 inter-station sections, wherein each section has 3 optimized alternative operation curves; respectively intercepting 35-minute passenger flow data of a peak, a flat peak and a night period in a day of a Fuzhou subway as passenger flow demands, and dividing the passenger flow data into 160 time nodes by taking 15s as a time interval; the arrival time boundaries of passengers at three stations S1, S2 and S3 are 35 minutes, 37 minutes and 40 minutes respectively, namely the arrival amount of each station passenger is zero after the corresponding boundaries are exceeded; consider a standby physical train of 10; consider random dynamic passenger flow data for 5 different scenarios.
(2) The minimum safety time interval H for two adjacent trains to arrive at or leave the same station is 3 minutes and 30 seconds, and the maximum time interval is used for avoiding overlong waiting timeThe maximum bearing capacity C max of the train is 1500, and the stop time of the train at each station is 30 seconds;
(3) Train mass M 0＝2.28×105 KG, average passenger mass M p =60 KG, passenger average energy consumption P ac =110W/passenger ·s in the car, and energy consumption P lc =50 KW/train of auxiliary equipment (air conditioner, lighting, etc.) in the car.
(4) Fixed cost (employee wages, equipment losses, etc.) Q 0＝2×104 RMB/train per active service performed by the train
(5) The operation time(s) and unit energy consumption (m 2/s2) corresponding to each inter-station operation curve are shown in table 1:
TABLE 1 basic parameters of the lines
(6) Scene probability α ω =0.2; according to the urban commercial electricity peak-valley time-sharing electricity price standard, namely peak electricity price: flat electricity price: night electricity price=2:3:2, three groups of weight coefficients in different time periods are set: peak time: β c＝1×104,βt＝3×104,βe = 2; flat period: β c＝1×104,βt＝3×104,βe = 3; night time period: β c＝1×104,βt＝2×104,βe = 2;
Based on the given parameters and data conditions, writing codes in MATLAB, constructing a model framework proposed by the method, and calling optimization software CPLEX to solve so as to obtain a corresponding train schedule; meanwhile, a variable neighborhood search algorithm is realized by using C++ programming to solve the time schedule problem, and train running diagrams of peak, flat peak and night time periods as shown in fig. 4,5 and 6 are respectively obtained after 208 seconds, 70 seconds and 17 seconds of calculation, wherein the rectangular height represents the waiting time of passengers arriving at the moment waiting for the current train at a platform. For the problem of robust optimization of train schedules in different time periods, the calculation results and the solving efficiency of the two methods are shown in table 2.
Table 2 comparison of calculated results
/>
Analysis of results:
As can be seen from the above results, the passenger demand is larger in peak hours, and the primary goal of the operation company is to effectively reduce the waiting time of the passengers, so that the effective service quantity should be increased, and the service train number should be increased; and the subway operation company in the peak-to-peak period can reduce the number of effective services to reduce the operation cost and the energy consumption cost. In three different time periods, the train is shown in the first station outgoing time interval pair such as shown in fig. 7, wherein the time interval between adjacent trains in the peak time period is smaller, and the time interval between adjacent trains is maximum in the night time period due to the small passenger flow.
In summary, the subway train schedule robust optimization method for random dynamic passenger flow and energy-saving operation disclosed by the invention establishes the association constraint between the effective service quantity, the speed curve selection decision variable and the train time voting decision variable, builds an integrated mathematical optimization model among the three, enhances the robustness of solving the mathematical model of the train schedule problem, and improves the problem of step-by-step independent optimization of different time schedules such as peak, flat peak, night and the like of the existing method. Meanwhile, the difference of the total mass of the train in different scenes is considered, and the running time of the train among stations is optimized by introducing a speed curve to select decision variables so as to reduce the energy consumption cost.
It should be understood that the foregoing examples of the present invention are provided merely for the purpose of clearly illustrating the present invention and are not intended to limit the embodiments of the present invention, and that various other changes and modifications may be made therein by one skilled in the art without departing from the spirit and scope of the present invention as defined by the appended claims.
What is not described in detail in this specification is prior art known to those skilled in the art.
Claims (3)
1. A robust optimization method for train schedules facing random dynamic passenger flow and energy-saving operation is characterized by comprising the following steps:
s1, recording station set of lines in a route as The station 1 is a starting station, and the station N is a destination station; record the set of available physical trains as/>A total of K trains can be put into service; recording the scene set of passenger flow as/>Record the discrete time period set/>Wherein t represents a t-th period;
Effective service decision variable y k, train departure time decision variable d 1,k at the first station and running curve selection decision variable among stations are introduced into robust optimization model of train schedule Where k represents train k,/>
S2, analyzing the relation between the train schedule and the departure time decision variable of the train at the head station and the operation curve selection decision variable among stations, and constructing linear association constraint between the train schedule and the departure time decision variable d i,k of the train and the operation curve selection decision variable among the stations;
s3, constructing nonlinear association constraint between the train schedule and an effective service decision variable y k; constructing train operation safety time interval constraint and train bearing capacity constraint;
S4, constructing a robust optimization model of a train schedule for random dynamic passenger flow and energy-saving operation, which aims at the fixed cost of the train, the total waiting time of passengers and the total energy consumption of the train, by combining the steps S1-S3, wherein decision variables of the robust optimization model comprise effective service decision variables, decision variables of departure time of the train at the departure time of the first station and decision variables of operation curves of the train among stations;
S5, solving the robust optimization model to obtain a global optimal solution serving as an optimal train schedule;
The specific steps of the step S2 are as follows: introducing a binary auxiliary variable z i,k(t),zi,k (t) =1 to indicate that the train k has arrived or passed the station i at the time t, and z i,k (t) =0 to indicate that the train k has not arrived and has not passed the station i at the time t, wherein the variable satisfies a non-increasing constraint as shown in the formula (2); the departure time d i,k of the train k at the station i is expressed as the expression (3):
Wherein, For station set in route,/>For the set of available physical trains,/>Is a set of discrete time periods; d i,k is a departure time decision variable of the train, which indicates that the train k starts from the station i at the time d i,k, a i+1,k indicates that the train arrives at the subsequent station i+1 at the time a i+1,k, and the running time r i,k of the interval [ i, i+1] is determined by the running curve selected when the train k runs on the interval; at this time, there is a linear association constraint between the train schedule and the departure time decision variable and the inter-station operation curve selection decision variable of the train as follows:
Wherein, For station set in route,/>For the set of available physical trains, a i,k is the time when train k arrives at station i, d i,k is the time when train k leaves station i, s i,k is the stop time of train k at station i, r i,k is the running time of train k in interval [ i, i+1 ]/>For all the set of alternative operating curves over interval [ i, i+1 ]/>The operation time corresponding to the operation curve f in the section [ i, i+1 ]; the formula (4) and the formula (5) are the departure and arrival time constraints of the train, the formula (6) is the association constraint of the decision variables selected by the running time between stations and the running curve between stations, and the formula (7) is the uniqueness constraint of the running curve selected by the train k when running on the interval [ i, i+1 ];
the step of constructing a nonlinear association constraint between the train schedule and the effective service decision variable y k in step S3 is: establishing a non-subtractive constraint for the effective service decision variable y k as shown in equation (8):
in order for a train executing effective service to complete the service within a planned time interval, the departure time of an effective terminal at each station should be later than the arrival time boundary of the passenger at the station, and the departure time of an effective next terminal at each station should be strictly earlier than the arrival time boundary of the passenger at the station; if train Executing effective service, recording y k =1, and adopting y k-yk+1 =1 to represent that the train k is an effective terminal bus, and adopting y k+1-yk+2 =1 to represent that the train k is an effective next terminal bus; the departure time constraint of the effective terminal bus at each station and the departure time constraint of the effective sub terminal bus at each station are respectively shown as the following formula (9) and the formula (10):
Wherein, A passenger arrival time boundary for station i; formulas (9) and (10) are nonlinear constraints;
The step of constructing the train operation safety time interval constraint in the step S3 is as follows: the departure time interval of the train k and the preceding train k-1 at the station i is recorded as h i,k,hi,k＝di,k-di,k-1, The train safety time interval constraint is constructed as follows:
Wherein H and The minimum interval time and the maximum interval time for two adjacent trains to leave or arrive at the station are respectively;
the step of constructing the bearing capacity constraint of the train in the step S3 is as follows: constructing a binary auxiliary variable L i,k(t),Li,k (t) =1 representing that a passenger of the station i can ride the train number k in the current t-th time period by the binary auxiliary variable z i,k (t) representing the train state in the step S2, wherein the t-th time period represents that the passenger reaching the station i can ride the train number k; l i,k (t) =0, indicating that the passenger arriving at station i in the t-th time period cannot ride the number k of cars; specifically expressed as formula (16):
Therefore, the number of passengers getting on and off the train k at the station i, the number of passengers in the carriage and the constraint of the train bearing capacity under the omega scene are constructed as follows:
Wherein, P i,j,ω (t) is the number of passengers arriving at station i and traveling to station j in the t-th time period in the omega scene, and P i′,i,ω (t) is the number of passengers arriving at station i' and traveling to station i in the t-th time period in the omega scene; in formula (17)For the number of boarding persons for train k at station i in ω scene,/>, in equation (18)For the number of passengers getting off the train k at the station i in the omega scene, C i,k,ω is the total number of passengers in a carriage when the train k leaves the station i in the omega scene, and C max is the maximum bearing capacity of the train; equation (19) describes the dynamic change process of the number of passengers in station i car for train k in ω scene, which must meet the maximum bearing capacity constraint (20) of the train to guarantee the feasibility of the current train schedule;
The specific steps of the step S4 are as follows:
1) The total number of effective train numbers running in the planning time interval is According to the effective service unit fixed cost Q 0, an objective function representing the total operation cost is established, as shown in the formula (21), and the total fixed cost of the train is as follows:
2) After the passengers enter the platform, waiting for the incoming trains in the waiting area, wherein the waiting time of the passengers which arrive at the station i in the t-th time period and leave by taking the train number k is d i,k -t, and according to the definition of the binary auxiliary variable L i,k (t), the expected value of the total waiting time of the passengers is calculated by the following formula:
Wherein TW ω is the total waiting time of passengers in the omega scene and alpha ω is the probability of scene omega;
3) The total energy consumption of the train comprises: traction energy consumption related to total mass of the train and auxiliary energy consumption related to the number of passengers and the inter-station travel time; the mass of the train in the empty state is recorded as M 0, and the average mass of passengers is recorded as M p; the unit traction energy consumption is u i,k when the train runs in an empty state, and is specifically expressed as a formula (26); traction energy consumption occurs if and only if the train performs an active service, and thus the traction energy consumption when the train k runs over the section [ i, i+1] in ω scene is expressed as the formula (27):
Wherein mu 1,μ2,μ3 is a constant parameter calculated from actual operation data;
Note that P ac is the average power consumption per unit time in the carriage, and P lc is the power consumption per unit time of auxiliary equipment in the carriage; likewise, auxiliary energy consumption occurs if and only if the train performs an active service, and thus, the auxiliary energy consumption when the train k runs on the section [ i, i+1] in the ω scene is represented by the formula (28):
then, the total energy consumed by the train running in ω scene is calculated by the equation (29):
Thus, the expected value of the total energy consumption of a scene-based train is:
the constructed robust optimization model for the train schedule facing the random dynamic passenger flow and energy-saving operation is as follows:
j is the total cost of the operator and the passenger in the planned time interval; FC is the total fixed cost of the train; TW ω is the total waiting time of the passengers in scene ω; e ω is the total energy consumption of the train in scene omega; alpha ω is scene probability; beta c,βt,βe is the weight coefficient of the fixed cost, the passenger waiting time cost and the train energy consumption cost, respectively.
2. The robust optimization method for train schedules for random dynamic passenger flow and energy-saving operation according to claim 1, wherein:
In step S1, the valid service decision variable y k =1 indicates that the train k performs a valid service, and y k =0 indicates that the train k performs an invalid service;
The departure time decision variable d 1,k of the train at the first stop represents the moment when the train k leaves the station 1;
Decision variable for inter-station operation curve selection The selection curve f is shown when the train k is operating in the interval i, i+1.
3. The robust optimization method for train schedules for random dynamic passenger flow and energy-saving operation according to claim 1, wherein:
in step S5, solving the robust optimization model by adopting a variable neighborhood search algorithm, wherein the method comprises the following specific steps: setting dithering operators I.e. randomly generating a new feasible solution, wherein/>In the form of a vector of decision variables y k,/>In the form of a vector of decision variables d 1,k; definition of neighborhood Structure/>Namely exchanging departure intervals of the train number k 1 and the preceding train k 1 -1 and the train number k 2 and the preceding train k 2 -1, wherein k 1,k2 is a randomly selected valid service train number; definition of neighborhood Structure/>I.e. changing the inter-station operating curve of each train number, where/>For decision variables/>Vector form of (a); definition of neighborhood Structure/> Regenerating a feasible departure time sequence/>, according to the number of effective service vehicles K p in the current solutionSelecting a feasible running curve for each train; thus, the form of the solution of the constructed robust optimization model can be expressed as/>Recording search times count=0 of failing to find a better solution, wherein the termination condition is that count is more than or equal to MAX iteration;
Step 1: generating an initial feasible solution as S 0, and letting the global optimal solution s=s 0;
step 2: according to the dithering operator M, the global optimal solution S is disturbed, and an initial local optimal solution S' =M (S) is generated;
Step 3: using the neighborhood structure N 1 to perform neighborhood search on the solution S ', if the neighbor solution S "=n 1 (S ') is found, and f (S") is less than f (S '), updating the locally optimal solution: s' =s ", and repeat Step3; otherwise, turning to Step 4;
Step 4: using the neighborhood structure N 2 to perform neighborhood search on the solution S ', if the neighbor solution S "=n 2 (S ') is found, and f (S") is less than f (S '), updating the locally optimal solution: s' =s ", and repeat Step3; otherwise, turning to Step 5;
Step 5: using a neighborhood structure N 3 to perform neighborhood search on the solution S ', if a neighbor solution S "=n 3 (S') is found and f (S ') < f (S'), updating the locally optimal solution: s' =s ", and repeat Step3; otherwise, turning to Step 6;
Step 6: comparing the local optimal solution S 'with the global optimal solution S, and if f (S') < f (S), updating the global optimal solution: s=s', and let count=0, go to Step 2; otherwise, let count=count+1, go to Step 7;
Step 7: if the termination condition count is not more than MAX iteration, turning to Step 2; otherwise, the algorithm is terminated, and a global optimal solution S is output.
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| CN112784204A (en) * | 2020-12-28 | 2021-05-11 | 北京交通大学 | Train schedule and passenger flow control robust optimization method facing uncertain demand |
| CN112918523B (en) * | 2021-02-25 | 2023-05-02 | 北京电子科技职业学院 | Passenger flow collaborative accurate control method for crowded subway line based on train schedule optimization |
| CN113592419B (en) * | 2021-05-31 | 2024-05-14 | 南京理工大学 | Rail transit speed and time table optimization method considering passenger flow and energy conservation |
| CN116384596B (en) * | 2023-06-05 | 2023-08-25 | 交控科技股份有限公司 | Train schedule optimization method and device, electronic equipment and medium |
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